It's a commonplace to state that while other sciences (like biology) may always need the newest books, we mathematicians also use to use older books. While this is a qualitative remark, I would like to get a quantitative result. So what are "old" books still used?

Coming from (algebraic) topology, the first things which come to my mind are the works by Milnor. Frequently used (also as a topic for seminars) are his Characteristic Classes (1974, but based on lectures from 1957), his Morse Theory (1963) and other books and articles by him from the mid sixties.

An older book, which is sometimes used, is Steenrod's The Topology of Fibre Bundles from 1951, but this feels a bit dated already. Books older than that in topology are usually only read for historic reasons.

As I have only very limited experience in other fields (except, perhaps, in algebraic geometry), my question is:

What are the oldest books regularly used in your field (and which don't feel "outdated")?

According to the Jahrbuch database, the first edition was in 1915.
Moreover, this 1915 edition was an extended version of a 1902 book,
by Whittaker alone.

The last revision was in 1927.
The book is still in print, and widely used, not only by mathematicians
but by physicists and engineers.
Soon we will celebrate the centenary... It has 1056 citations on Mathscinet, by the way, and 8866 on the Google Scholar !

Perhaps this deserves a Guinnes book of records entry as a "textbook longest continuously in print".
And I suppose this is a record not only for math but for all sciences...
with the exception of Euclid and Ptolemy, of course:-)

If we include not only textbooks but research monographs there are plenty of other examples, even
older ones:

H. F. Baker, Abelian functions, was first published in 1897. Reprinted in 1995, and there is a new
Russian translation.

Just out of curiosity, look at its current citation rate in Mathscinet:-)

They also reprinted

H. Schubert, Kalkül der abzählenden Geometrie, 1879, in 1979,

and again you can see from Mathscinet
that people are using this.

EDIT: A brief inspection of the most cited (and thus most used) books on Mathscinet shows that
a very large proportion of the most cited books are 30-40 years old.
Which is easy to explain, by the way. Thus on my opinion, such books do not qualify for this list
(unless we want to make it infinite).

are checked out from my university library.
Mathscinet shows 1157 citations for the last 2 editions.

Another question is old papers which are still highly sited. A typical life span of a paper is much
smaller than that of a book. In the list of 100 most cited papers in 2011, I found only two papers
published before 1950 (One by Shannon and another by Leray).

I have an electronic copy of the 1996 reissue of Whittaker and Watson's that details its history: first edition 1902, second edition 1915, third edition 1920, fourth edition 1927. Since then, there were 8 reprints (1935, 1940, 1946, 1950, 1952, 1958, 1962 and 1963) and the 1996 reissue.
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Alberto García-RabosoDec 28 '12 at 19:25

@ayanta: Well, the new chapter on elliptic curves was written with an eye towards fitting into the style of the rest of the text. (An assertion that I feel that I'm entitled to state as a fact, rather than as an opinion.) So I guess there might be some who would say that the elliptic curves chapter is also "outdated", despite having been written quite recently! But I have to respectively disagree with your opinion of the book, which I feel is a masterpiece.
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Joe SilvermanDec 29 '12 at 23:37

Henri Cartan and Samuel Eilenberg published their Homological Algebra in 1956, although it was famously circulated for a long time before that. While that book more or less founded its subject, it is still quite useful.

Has anyone else ever noticed something funny about the title of Chapter 1 in that edition?
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Adam EpsteinDec 29 '12 at 19:44

1

My own copy has COMPUTER NUMBERS. (Rodrigo, perhaps I showed you when you were in my lecture course?) But I would actually imagine that far more people (not necessarily mathematicians) would find COMPLEX NUMBERS funnier.
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Adam EpsteinJan 4 '13 at 14:07

Abramowitz and Stegun's Handbook of Mathematical Functions (1964) is still used. As the August 2011 Notices article by Boisvert et al. says,

The Handbook remains highly relevant today
in spite of its age. In 2009, for example, the Web
of Science records more than 2,000 citations to
the Handbook. That is more than one published
paper every five hours—quite remarkable!

The notes of the 1951-2 Artin-Tate seminar on class field theory (published in 1968, and re-issued in LaTeX form a few years ago with a new Introduction by Tate addressing subsequent developments) remains a fundamental reference in algebraic number theory, despite the abundant supply of more recent references on the subject.

One reason is that it is the only reference outside the original research literature where one can find a complete treatment (with proofs) of certain key aspects of the theory such as the Grunwald-Wang phenomenon and Weil groups for class formations (especially the case of number fields, which lacks a bare-hands construction as for local fields and global function fields). Come to think of it, the general notion of Weil groups for class formations emerged from that seminar...The style of the proofs remains generally quite fresh.

The standard, go to reference in geometric measure theory is still Federer's 1969 classic, Geometric Measure Theory. It is very rarely the first reference one uses since it is rather dense and there are other introductions and expositions, some of them very good.

For Salmon's book, the 4th edition of 1885 might be best. Indeed, as I learned from a paper by Macauley, it has a discussion (on p. 87) of Cayley's very general formula for the multivariate resultant as the determinant of a complex (see the book by Gelfand, Kapranov and Zelevinsky for a modern account and a reprint of Cayley's paper).

Since you answered before this was turned into CW by the questioner, your answer stayed in normal mode. Typically moderators would take care of this, but since your answer is the only one affected in this case, I thought it could be more efficient if you turned your answer into CW manually (edit and tick the box).
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quidDec 28 '12 at 18:18

@Abdelmalek Abdesselam: Can it really be that modern books on computational commutative algebra have not adequately replaced the need to look at a book on "modern higher algebra" from 1876 (or some of the others that you list)? This sounds very surprising. What are examples of things found in such old books that are not available in more recent references?
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user30180Dec 29 '12 at 6:00

3

@Ayanta: Despite the eloquence of your rethorical question, what you said is simply wrong. For instance anything involving the classical symbolic method in relation with specific invariants coming from elimination theory is not really accounted for nor "adequately replaced" in the recent commutative algebra literature. To form an accurate and informed opinion you need to have a look at the books I mentioned especially Grace and Young if you only have time to look at one.
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Abdelmalek AbdesselamDec 31 '12 at 12:31

Gaston Darboux' magnum opus Leçons sur la Théorie générale des Surfaces et les Applications géométriques du Calcul infinitésimal (first edition 1890, I think; there is a second edition dating from around 1915) is still read by many differential geometers, and, as far as I know, it is still in print via the AMS Chelsea series.

I used G. H. Hardy's A Course of Pure Mathematics (First edition 1908) when I taught undergraduate real analysis not so long ago. The care with which concepts are explained and the number of interesting problems and examples is, in my opinion, unmatched by newer books.

My own field, ergodic theory, is relatively young in that some concepts now regarded as fundamental -- Kolmogorov-Sinai entropy, for example -- were not fully formulated until around 1960. Nonetheless there are a couple of old books still in use and receiving citations:

E. Hopf, Ergodentheorie, 1937;

R. Halmos, Ergodic theory, 1957.

If the 1960s are sufficiently long ago to constitute "old" then there are many old references in probability which remain in heavy use, for example:

P. Billingsley, Convergence of probability measures, 1968;

L. Breiman, Probability, 1968;

and one of the classics of the field:

W. Feller, Introduction to probability theory and its applications, 1950.

Outside my own field, a much-cited number theory text which no-one has yet mentioned:

Montgomery and Zippin "Topological Transformation Groups" (originally published in 1955) is still the only book to cover the relevant results on topological characterization of Lie groups in full generality (including Lie group actions). I am not sure if this belongs to algebra or topology area-wise, but it is used in my area, geometric group theory.

For pedagogical purposes, I still use "What Is Mathematics?" by Courant and Robbins (originally published in 1941) and "Geometry and Imagination" (1932) by Hilbert and Kohn-Vossen, when a high school student or an undergraduate asks me for suggestions.

My personal definition of an "old book" is the same as Lee Mosher's, so I do not include here Chapters 4-6 of Bourbaki's "Lie groups and Lie algebras" (1968) which I use as a working tool.